Indeed, it is a known erratum of this book (see its website in the errata .pdf), that the specific lemma does not state the moment-boundedness condition $$\exists \; \delta : E(|z_n^{s+\delta}|) < M < \infty\;\; \forall n$$$$\exists \; \delta : E(|z_n|^{s+\delta}) < M < \infty\;\; \forall n$$
